Finite Group Databasebeta

← C23×#3805 #11919203

Subgroups — C23×#3805 #11919203

Order 1152.

13721 conjugacy classes of subgroups; only the 3844 normal subgroups are shown.

normal subgroups only

This group has 13721 conjugacy classes of subgroups — too many to draw. Only its 3844 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C23×C3×#67C212yesmaximal, complemented
3C23×#627C211yesmaximal, characteristic, complemented
4C23×#625C211yesmaximal, characteristic, complemented
5C22×#3805C2156yesmaximal, direct product factor
6C23×D3×Dic3C212yesmaximal, complemented
7C23×#237—32nomaximal
8C25×C32C2211yescharacteristic
9C24×C3×D3C2212yes—
10C23×C3⋊Dic3C2211yescharacteristic
11C23×C3⋊Dic3C2211yescharacteristic
12C23×C3⋊Dic3C22128yescomplemented
13C22×C3×#67C22156yescomplemented
14C23×C3×Dic3C2212yes—
15C22×#627C22128yescomplemented
16C22×#625C22128yescomplemented
17C2×#3805C221448yesdirect product factor
18C22×D3×Dic3C22156yescomplemented
19C23×C3×D4—32no—
20C23×#67—32no—
21C23×#67D312yescomplemented
22C24×Dic3—32no—
23C24×Dic3—32no—
24C24×Dic3—61no—
25C22×#237—3112no—
26C23×C4×D3—32no—
27C23×Dic6—32no—
28C24×C32—28no—
29C24×C32C23114yes—
30C24×C32C2311yescharacteristic
31C23×C3×D3—216no—
32C23×C3×D3C23128yes—
33C22×C3⋊Dic3C23114yes—
34C22×C3⋊Dic3C23114yes—
35C22×C3⋊Dic3C231112yescomplemented
36C2×C3×#67C231224yescomplemented
37C22×C3×Dic3C23128yes—
38C2×#627C231112yescomplemented
39C2×#625C231112yescomplemented
40#3805C231512yesdirect product factor
41C2×D3×Dic3C231224yescomplemented
42C23×#41—91noSylow
43C25×C3—21no—
44C25×C3—32no—
45C25×C3C2×D312yes—
46C24×D3C2×D312yes—
47C22×C3×D4—356no—
48C23×C3×C4—32no—
49C22×#67—356no—
50C22×#67C2×D3156yescomplemented
51C23×Dic3—32no—
52C23×Dic3—32no—
53C23×Dic3—356no—
54C23×Dic3—356no—
55C23×Dic3—61no—
56C23×Dic3—61no—
57C23×Dic3—628no—
58C23×Dic3C2×D312yes—
59C2×#237—3896no—
60C22×C4×D3—356no—
61C22×Dic6—356no—
62C23×C32—256no—
63C23×C32#4118yescomplemented
64C23×C32C24128yes—
65C23×C32C2417yes—
66C22×C3×D3—2112no—
67C22×C3×D3C24156yes—
68C2×C3⋊Dic3C24128yes—
69C2×C3⋊Dic3C24128yes—
70C2×C3⋊Dic3C24164yescomplemented
71C3×#67C241128yescomplemented
72C2×C3×Dic3C24156yes—
73#627C24164yescomplemented
74#625C24164yescomplemented
75D3×Dic3C241128yescomplemented
76C22×#41—956no—
77C23×Q8—91no—
78C23×D4—32no—
79C23×D4—91no—
80C24×C4—92no—
81C24×C4—91no—
82C24×C3—216no—
83C24×C3—216no—
84C24×C3—214no—
85C24×C3—21no—
86C24×C3—328no—
87C24×C3—616no—
88C24×C3C22×D3128yes—
89C24×C3C22×D312yes—
90C23×D3—216no—
91C23×D3C22×D3128yes—
92C2×C3×D4—3224no—
93C22×C3×C4—328no—
94C2×#67—3224no—
95C2×#67C22×D31224yescomplemented
96C22×Dic3—328no—
97C22×Dic3—328no—
98C22×Dic3—3224no—
99C22×Dic3—3224no—
100C22×Dic3—614no—
101C22×Dic3—614no—
102C22×Dic3—6112no—
103C22×Dic3C22×D3128yes—
104#237—31024no—
105C2×C4×D3—3224no—
106C2×Dic6—3224no—
107C22×C32—256no—
108C22×C32C2×#41128yescomplemented
109C22×C32C2518yes—
110C22×C32C2517yes—
111C2×C3×D3—2112no—
112C2×C3×D3C25116yes—
113C3⋊Dic3C2518yes—
114C3⋊Dic3C2518yes—
115C3×Dic3C25116yes—
116C25—32no—
117C25D3211yescharacteristic
118C2×#41—9448no—
119C22×Q8—928no—
120C22×D4—356no—
121C22×D4—928no—
122C23×C4—32no—
123C23×C4—956no—
124C23×C4—928no—
125C23×C4—91no—
126C23×C4—91no—
127C23×C3—2112no—
128C23×C3—2112no—
129C23×C3—228no—
130C23×C3—27no—
131C23×C3—28no—
132C23×C3—356no—
133C23×C3—6112no—
134C23×C3#237116yescomplemented
135C23×C3C23×D3156yes—
136C23×C3C23×D3114yes—
137C22×D3—2112no—
138C22×D3C23×D3156yes—
139C3×D4—3128no—
140C2×C3×C4—356no—
141#67—3128no—
142#67C23×D31128yescomplemented
143C2×Dic3—356no—
144C2×Dic3—356no—
145C2×Dic3—3128no—
146C2×Dic3—3128no—
147C2×Dic3—628no—
148C2×Dic3—628no—
149C2×Dic3—664no—
150C2×Dic3C23×D3156yes—
151C4×D3—3128no—
152Dic6—3128no—
153C2×C32—28no—
154C2×C32C22×#41114yescomplemented
155C2×C32C2611yescharacteristic
156C3×D3—216no—
157C24—28no—
158C24—328no—
159C24—616no—
160C24C2×D32114yes—
161C24C2×D3211yescharacteristic
162#41—9512no—
163C2×Q8—9112no—
164C2×D4—3224no—
165C2×D4—9112no—
166C22×C4—328no—
167C22×C4—9224no—
168C22×C4—914no—
169C22×C4—914no—
170C22×C4—9112no—
171C22×C3—2112no—
172C22×C3—2112no—
173C22×C3—228no—
174C22×C3—28no—
175C22×C3—27no—
176C22×C3—316no—
177C22×C3—6112no—
178C22×C3C2×#237156yescomplemented
179C22×C3C24×D3116yes—
180C22×C3C24×D3114yes—
181C2×D3—2112no—
182C2×D3C24×D3116yes—
183C3×C4—316no—
184Dic3—316no—
185Dic3—316no—
186Dic3—68no—
187Dic3—68no—
188Dic3C24×D3116yes—
189C32C23×#4111yescharacteristic, complemented, Sylow
190C23—256no—
191C23—356no—
192C23—6112no—
193C23#380518yesdirect product factor
194C23C22×D32128yes—
195C23C22×D3217yes—
196Q8—964no—
197D4—3128no—
198D4—964no—
199C2×C4—356no—
200C2×C4—9128no—
201C2×C4—928no—
202C2×C4—928no—
203C2×C4—964no—
204C2×C3—216no—
205C2×C3—216no—
206C2×C3—214no—
207C2×C3—21no—
208C2×C3—616no—
209C2×C3C22×#237128yescomplemented
210C2×C3C25×D312yes—
211D3—216no—
212C22—256no—
213C22—316no—
214C22—6112no—
215C22C2×#3805128yesdirect product factor
216C22C23×D3218yes—
217C22C23×D3217yes—
218C4—316no—
219C4—98no—
220C4—98no—
221C3—21no—
222C3C23×#23712yescomplemented
223C2—28no—
224C2—616no—
225C2C22×#3805114yesdirect product factor
226C2C24×D3211yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC2×C32C26
FrattiniC2C24×D32
FittingC25×C32C22
socleC24×C32C23
hypercenterC25D32
hypocenterC32C23×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24 ◁ C25

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C24×C3⋊Dic3C2semidirect
2. C23×C3×#67C2semidirect
3. C23×#627C2semidirect
4. C23×#625C2semidirect
5. C22×#3805C2direct (N × H)
6. C23×D3×Dic3C2semidirect
12. C23×C3⋊Dic3C22semidirect
13. C22×C3×#67C22semidirect
15. C22×#627C22semidirect
16. C22×#625C22semidirect
17. C2×#3805C22direct (N × H)
18. C22×D3×Dic3C22semidirect
21. C23×#67D3semidirect
35. C22×C3⋊Dic3C23semidirect
36. C2×C3×#67C23semidirect
38. C2×#627C23semidirect
39. C2×#625C23semidirect
40. #3805C23direct (N × H)
41. C2×D3×Dic3C23semidirect
50. C22×#67C2×D3semidirect
63. C23×C32#41semidirect
70. C2×C3⋊Dic3C24semidirect
71. C3×#67C24semidirect
73. #627C24semidirect
74. #625C24semidirect
75. D3×Dic3C24semidirect
95. C2×#67C22×D3semidirect
108. C22×C32C2×#41semidirect
134. C23×C3#237semidirect
142. #67C23×D3semidirect
154. C2×C32C22×#41semidirect
178. C22×C3C2×#237semidirect
189. C32C23×#41semidirect
193. C23#3805direct (N × H)
209. C2×C3C22×#237semidirect
215. C22C2×#3805direct (N × H)
222. C3C23×#237semidirect
225. C2C22×#3805direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C24×C3⋊Dic32
C23×C3×#672
C23×#6272
C23×#6252
C22×#38052
C23×D3×Dic32
C23×#2373